The remainder when (2023)²⁰²³ is divided by 35 is __________.

Solution

Maths sir · 3 steps

Write 2023 as 2030 - 7. Since 2030 is a multiple of 35, only the last binomial term matters.

1. Split near 35. 2030 = 35 × 58, so 2023 = 35 × 58 - 7. Expand (35 × 58 - 7)²⁰²³ binomially. Every term except the last has a factor of 35, and the last term is (-7)²⁰²³ = -7²⁰²³ because the power is odd.

   2023²⁰²³ = 35k - 7²⁰²³

2. Handle the leftover. 7²⁰²³ = 7 × 49¹⁰¹¹ = 7(50 - 1)¹⁰¹¹. Since 1011 is odd, (50 - 1)¹⁰¹¹ = 50m - 1, so 7²⁰²³ = 350m - 7. Note that 350 = 35 × 10.
3. So. 2023²⁰²³ = 35k - 350m + 7 = 35(k - 10m) + 7. The remainder is 7.

The common slip: 28. Dropping the minus sign in (-7)²⁰²³ leads to 28, not 7, because 7²⁰²³ by itself leaves remainder 28 on division by 35.

The idea to keep. To find the remainder of aⁿ on division by d, write a as a multiple of d plus or minus a small number. Only the last term survives. Repeat the trick on that term, and watch the sign.

Answer from the NTA final answer key. Solution by PrepWiser.